Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > mvrcl | Structured version Visualization version GIF version |
Description: A power series variable is a polynomial. (Contributed by Mario Carneiro, 9-Jan-2015.) |
Ref | Expression |
---|---|
mvrcl.s | ⊢ 𝑃 = (𝐼 mPoly 𝑅) |
mvrcl.v | ⊢ 𝑉 = (𝐼 mVar 𝑅) |
mvrcl.b | ⊢ 𝐵 = (Base‘𝑃) |
mvrcl.i | ⊢ (𝜑 → 𝐼 ∈ 𝑊) |
mvrcl.r | ⊢ (𝜑 → 𝑅 ∈ Ring) |
mvrcl.x | ⊢ (𝜑 → 𝑋 ∈ 𝐼) |
Ref | Expression |
---|---|
mvrcl | ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (𝐼 mPwSer 𝑅) = (𝐼 mPwSer 𝑅) | |
2 | mvrcl.v | . . 3 ⊢ 𝑉 = (𝐼 mVar 𝑅) | |
3 | eqid 2738 | . . 3 ⊢ (Base‘(𝐼 mPwSer 𝑅)) = (Base‘(𝐼 mPwSer 𝑅)) | |
4 | mvrcl.i | . . 3 ⊢ (𝜑 → 𝐼 ∈ 𝑊) | |
5 | mvrcl.r | . . 3 ⊢ (𝜑 → 𝑅 ∈ Ring) | |
6 | mvrcl.x | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐼) | |
7 | 1, 2, 3, 4, 5, 6 | mvrcl2 20805 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅))) |
8 | fvexd 6689 | . . 3 ⊢ (𝜑 → (𝑉‘𝑋) ∈ V) | |
9 | eqid 2738 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
10 | eqid 2738 | . . . . 5 ⊢ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} = {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} | |
11 | 1, 9, 10, 3, 7 | psrelbas 20758 | . . . 4 ⊢ (𝜑 → (𝑉‘𝑋):{𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}⟶(Base‘𝑅)) |
12 | 11 | ffund 6508 | . . 3 ⊢ (𝜑 → Fun (𝑉‘𝑋)) |
13 | fvexd 6689 | . . 3 ⊢ (𝜑 → (0g‘𝑅) ∈ V) | |
14 | snfi 8642 | . . . 4 ⊢ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin | |
15 | 14 | a1i 11 | . . 3 ⊢ (𝜑 → {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin) |
16 | eqid 2738 | . . . . . 6 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
17 | eqid 2738 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
18 | 4 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝐼 ∈ 𝑊) |
19 | 5 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑅 ∈ Ring) |
20 | 6 | adantr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑋 ∈ 𝐼) |
21 | simpr 488 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) | |
22 | eldifsn 4675 | . . . . . . . 8 ⊢ (𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) ↔ (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) | |
23 | 21, 22 | sylib 221 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∧ 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)))) |
24 | 23 | simpld 498 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ∈ {𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin}) |
25 | 2, 10, 16, 17, 18, 19, 20, 24 | mvrval2 20801 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅))) |
26 | 23 | simprd 499 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → 𝑥 ≠ (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
27 | 26 | neneqd 2939 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ¬ 𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))) |
28 | 27 | iffalsed 4425 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → if(𝑥 = (𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0)), (1r‘𝑅), (0g‘𝑅)) = (0g‘𝑅)) |
29 | 25, 28 | eqtrd 2773 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ({𝑓 ∈ (ℕ0 ↑m 𝐼) ∣ (◡𝑓 “ ℕ) ∈ Fin} ∖ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → ((𝑉‘𝑋)‘𝑥) = (0g‘𝑅)) |
30 | 11, 29 | suppss 7889 | . . 3 ⊢ (𝜑 → ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))}) |
31 | suppssfifsupp 8921 | . . 3 ⊢ ((((𝑉‘𝑋) ∈ V ∧ Fun (𝑉‘𝑋) ∧ (0g‘𝑅) ∈ V) ∧ ({(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))} ∈ Fin ∧ ((𝑉‘𝑋) supp (0g‘𝑅)) ⊆ {(𝑦 ∈ 𝐼 ↦ if(𝑦 = 𝑋, 1, 0))})) → (𝑉‘𝑋) finSupp (0g‘𝑅)) | |
32 | 8, 12, 13, 15, 30, 31 | syl32anc 1379 | . 2 ⊢ (𝜑 → (𝑉‘𝑋) finSupp (0g‘𝑅)) |
33 | mvrcl.s | . . 3 ⊢ 𝑃 = (𝐼 mPoly 𝑅) | |
34 | mvrcl.b | . . 3 ⊢ 𝐵 = (Base‘𝑃) | |
35 | 33, 1, 3, 16, 34 | mplelbas 20809 | . 2 ⊢ ((𝑉‘𝑋) ∈ 𝐵 ↔ ((𝑉‘𝑋) ∈ (Base‘(𝐼 mPwSer 𝑅)) ∧ (𝑉‘𝑋) finSupp (0g‘𝑅))) |
36 | 7, 32, 35 | sylanbrc 586 | 1 ⊢ (𝜑 → (𝑉‘𝑋) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ≠ wne 2934 {crab 3057 Vcvv 3398 ∖ cdif 3840 ⊆ wss 3843 ifcif 4414 {csn 4516 class class class wbr 5030 ↦ cmpt 5110 ◡ccnv 5524 “ cima 5528 Fun wfun 6333 ‘cfv 6339 (class class class)co 7170 supp csupp 7856 ↑m cmap 8437 Fincfn 8555 finSupp cfsupp 8906 0cc0 10615 1c1 10616 ℕcn 11716 ℕ0cn0 11976 Basecbs 16586 0gc0g 16816 1rcur 19370 Ringcrg 19416 mPwSer cmps 20717 mVar cmvr 20718 mPoly cmpl 20719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-of 7425 df-om 7600 df-1st 7714 df-2nd 7715 df-supp 7857 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-1o 8131 df-er 8320 df-map 8439 df-en 8556 df-dom 8557 df-sdom 8558 df-fin 8559 df-fsupp 8907 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-3 11780 df-4 11781 df-5 11782 df-6 11783 df-7 11784 df-8 11785 df-9 11786 df-n0 11977 df-z 12063 df-uz 12325 df-fz 12982 df-struct 16588 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-mulr 16682 df-sca 16684 df-vsca 16685 df-tset 16687 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-mgp 19359 df-ur 19371 df-ring 19418 df-psr 20722 df-mvr 20723 df-mpl 20724 |
This theorem is referenced by: subrgmvrf 20845 mplcoe3 20849 mplcoe5lem 20850 mplcoe5 20851 mplcoe2 20852 mplbas2 20853 mvrf2 20872 evlsvarpw 20908 mpfproj 20916 mpfind 20921 mhpvarcl 20942 vr1cl 20992 selvval2lem5 39811 evlsvarval 39853 |
Copyright terms: Public domain | W3C validator |